Remark 4.3 Consider the level \(p\)  congruence subgroup \(\operatorname{Sp}_{2n}({\mathbb {Z}})[p]\) ,
which is the kernel of the homomorphism \(\operatorname{Sp}_{2n}({\mathbb {Z}}) \rightarrow \operatorname{Sp}_{2n}({\mathbb {F}}_p)\)  induced by reduction modulo \(p\) .
By [1]} it is torsion-free when \(p\ne 2\) .
Let \(\operatorname{St}(C_n,{\mathbb {Z}})[p]\)  be the kernel of the composite map
\(\operatorname{St}(C_n,{\mathbb {Z}}) \overset{\pi }{\longrightarrow } \operatorname{Sp}_{2n}({\mathbb {Z}}) \rightarrow \operatorname{Sp}_{2n}({\mathbb {F}}_p)\) ;
it is of finite index in \(\operatorname{St}(C_n,{\mathbb {Z}})\) .
Now \(\operatorname{St}(C_n,{\mathbb {Z}})[p]\)  is an extension of the torsion-free group \(\operatorname{Sp}_{2n}({\mathbb {Z}})[p]\)  by the infinite cyclic group \(\langle w_1^4 \rangle \) ;
therefore it is torsion-free as well (\(p\ne 2\) ).
It follows that the Steinberg group \(\operatorname{St}(C_n,{\mathbb {Z}})\)  is virtually torsion-free, i.e. has finite-index torsion-free subgroups.

